Integral representations for a class of triple confluent hypergeometric functions and their applications in boundary value problems

Hypergeometric functions are divided into complete and confluent functions. Srivastava and Karlsson were the first to propose a method for constructing the complete set of triple Gaussian hypergeometric series and compiled a table containing definitions and regions of convergence for 205 distinct complete series in three variables. Subsequently, several authors obtained various integral representations and transformation formulas for the functions introduced by Srivastava and Karlsson. More recently, Ergashev identified 395 hypergeometric series of three variables that represent confluent forms of the known 205 complete hypergeometric series. In the present study, new Euler-type integral representations are derived for certain Gaussian hypergeometric functions of three variables. The main results are obtained using properties of the gamma and beta functions. New integral representations are established for 14 functions from the list of confluent hypergeometric functions of three variables. All derived integrals can be regarded as generalized Euler type representations of the classical Gaussian hypergeometric functions of one and two variables. In addition, it is demonstrated how one of these confluent functions, together with its integral representation, can be applied to construct solutions of the three-dimensional singular Helmholtz equation.